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Chapter 3: Q45E (page 132)

Question: Are the columns of an invertible matrix linearly independent?

Short Answer

Expert verified

Yes, the columns of an invertible matrix are linearly independent.

Step by step solution

01

Definition of Linearly independent

Linearly independent is defined as the property of a set having no linear combination of its elements equal to zero when the coefficients are taken from a given set unless the coefficient of each element is zero.

02

Kernel of invertible

For vectors v1鈫�,v1鈫�,...,vm鈫�,, we know that the following statements are equivalent -\par1. The vectors v1鈫�,v1鈫�,...,vm鈫�,are linearly independent -/par2.kerA=0, where is the given vectors as their columns.

Also, the kernel of an invertible matrix is zero.

03

The final answer

A is invertible \par鈬�kerA=0\par鈬�Column vectors of are linearly independent.

Therefore, the columns of an invertible matrix are linearly independent.

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Most popular questions from this chapter

Give an example of a matrixAsuch thatim(A)is spanned by the vector[15].

Determine whether the following vectors form a basis of 鈩�4; [1111],[1-11-1],[1248],[1-24-8].

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Question: Consider three linearly independent vectorsv1鈫�,v2鈫�,v3鈫�in 鈩�3. Are the vectorsv1鈫�,v1鈫�+v2鈫�,v1鈫�+v2鈫�+v3鈫�linearly independent as well? How can you tell?

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